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BY THE SAME A 
The Golden Person in the 


Episodes from an Unw 
A Primer of Higher Spac 
Four-Dimensional Vistas. 


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“ORNAMENT 


Claude on 


EW_YOR 
ALFRED A’ KNOPF 
1927 , 


Copyright 1915 by Claude Bragd , 


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DEDICATED TO E. B. 


CONTENTS 


THE NEED OF A NEW FORM LANGUAGE 
ORNAMENT AND PSYCHOLOGY 

THE KEY TO PROJECTIVE ORNAMENT 
THREE REGULAR POLYHEDROIDS 
FOLDING DOWN - 

MAGIC LINES IN MAGIC SQUARES 

A PHILOSOPHY OF ORNAMENT 


THE USES OF PROJECTIVE ORNAMENT 


FOREWORD 


ANY sincere workers in the field of art have 

realized the aesthetic poverty into which the 
modern world has fallen. Designers are reduced 
either to dig in the boneyard of dead civilizations, 
or to develop a purely personal style and method. 
The latter is rarely successful: city dwellers that we 
are for the most part, and self-divorced from Nature, 
she witholds her intimate secrets from us. Our 
ignorance and superficiality stand pitifully revealed. 


Is there not some source, some secret spring of 
fresh beauty undiscovered, to satisfy our thirsty 
souls? Having all his life asked himself this ques- 
tion, the author at last undertook its quest. Such 
results as have up to the present rewarded his 
search are here set forth. Their value and import- 
ance will be determined, as all things are determined, 
by use and time, but this much must be admitted— 
they are drawn from a deep well. 


The author desires to acknowledge his indebted- 
ness to the following sources for material contained 
in this volume: The Fourth Dimension, by C. 
Howard Hinton, M. A.; Geometry of Four Dimen- 
sions, by Henry Parker Manning, Ph. D.; Obser- 
vational Geometry, by William T. Campbell, A. M.; 
Mathematical Essays and Recreations, by Hermann 


PROJECTIVE ORNZAARE 


Schubert; also to an essay entitled Regular Figures in 
n-dimensional Space, by W. I. Stringham, in the third 
volume of the American Journal of Mathematics, 
and an article on Magic Squares in the Eleventh 
Edition of the Encyclopaedia Britannica. 

The chapter entitled 4 Philosophy of Ornament 
is enriched by certain ideas first suggested in a lec- 
ture by Mr. Irving K. Pond. With no desire to 
wear borrowed plumes, the author yet found it im- 
possible in this instance to avoid doing so, they are 
so woven into the very texture of his thought. In 
the circumstances he can only make grateful acknow- - 
ledgement to Mr. Pond. 

The author desires to express his gratitude to Mr. 
Frederick L. Trautmann for his admirable inter- 
pretations of Projective Ornament in color, of which 
the frontispiece gives an idea—and only an idea. 


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THE NEED OF A NEW FORM 
LANGUAGE 


We are without a form language suitable to the needs of today. Archi- 
tecture and ornament constitute such a language. Structural necessity 
may be depended upon to evolve fit and expressive architectural forms, 
but the same thing is not true of ornament. This necessary element 
might be supplied by an individual genius, it might be derived from 
the conventionalization of natural forms, or lastly it might be devel- 
oped from geometry. The geometric source is richest in promise. 


ARCHITECTURE AND ORNAMENT 


N contemplating the surviving relics of any period 
in which the soul of a people achieved aesthetic 
utterance through the arts of space, it is clear that 
in their architecture and in their ornament they had 
a form language as distinctive and adequate as any 
spoken language. Today we have no such language. 
This is equivalent to saying that we have not at- 
tained to aesthetic utterance through the arts of 
space. That we shall attain to it, that we shall 
develop a new form language, it is impossible to 
doubt; but not until after we realize our need, and 
set about supplying it. 


PROJECTIVE ORNAZARe 


Consider the present status of architecture, 
which is preéminently the art of space. Modern 
architecture, except on its en- 
gineering side, has not yet 
found itself: the style of a 
building is determined, not 
by necessity, but by the whim 
of the designer; it is made up 
of borrowings and survivals. 
So urgent is the need of more 
appropriate and indigenous 
architectural forms with 
which to clothe the steel 
framework for which some 
sort of protective covering is 
of first importance, that some 
architects have ceased search- 
ing in the cemetery of a too 

Pontshatenids sacredly cherished past. They 

are seeking to solve their 

problems rather by a process of elimination, using 

the most elementary forms and the materials readiest 

to hand. In thus facing their difficulty they are re- 
creating their chosen art, and not abrogating it. 


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The development of new architectural forms 
appropriate to the new structural methods is already 
under way, and its successful issue may safely be 
left to necessity and to time; but the no less urgent 
need of fresh motifs in ornament has not: yet even 
begun to be met. So far as architecture is con- 
cerned, the need is acute only for those who are 
determined to be modern. Having perforce abandon- 
ed the structural methods of the past, and the forms 

2 


PROJECTIVE ORNAMENT 


associated with these methods, they nevertheless 
continue to use the ornament associated with what 
they have abandoned: the clothes are new, but not 
the collar and necktie. The reason for this failure 
of invention is that while common sense, and a 
feeling for fitness and proportion, serve to produce 
the clothing of a building, the faculty for originating 
appropriate and beautiful ornament is one of the 
rarest in the whole range of art. Those arts of space 
which involve the element of decoration suffer 
from the same lack, and for a similar reason. 


Three possible sources of supply suggest them- 
selves for this needed element in a new form language. 
Ornament might be the single-handed creation of an 
original genius in this partic- 
ular field; it might be de- 
rived from the conventional- 
ization of native flora, as it 
was in the past; or it might 
be developed from geometry. 
Let us examine each of these 
possibilities in turn. 


The first we must reject. ch 
Even supposing that this art Pegt 
saviour should appear as some 
rarely gifted and resourceful 
creator of ornament, it would 
be calamitous to impose the 
idiosyncratic space rhythm of 
a single individual upon an 
entire architecture. Fortu- Tesseracts: Cubes 
nately such a thing is impossible. In Mr. Louis 
Sullivan, for example, we have an ornamentalist 

3 


PROJECTIVE ORNAMEW® 


of the highest distinction (quite aside from his 
sterling qualities as an architect), but from the 
work of his imitators it is 
clear that his secret is in- 
communicable. It would be 
better for his disciples to de- 
velop an individual manner 
of their own, and this a few 
of them are doing. Mr. 
Sullivan will leave his little 
legacy of beauty for the en- 
richment of those who come 
after, but our hope for an 
ornament less personal, more 
universal and generic, will be 
as far from realization as 
before. 


NATURE 


Tetrahedrons: Tesseracts: : ‘ 
Icositetrahedroid Such a saviour being by 


the very necessities of the 
case ‘denied, us, may, we not go directly to Nature 
and choose whatever patterns suit our fancy from 
the rich garment which she weaves and wears? 
There is no lack of precedent for such a procedure. 
The Egyptian lotus, the Greek honeysuckle, the 
acanthus, the Indian palmette, achieved, in this way, 
their apotheosis in art. The Japanese use their 
chrysanthemum, their wisteria and bamboo, in 
similar fashion; so why may not we do likewise? 
The thing has already been attempted, but never 
consistently nor successfully. 
While far from solving the problem of a new 
language of ornament, for reasons presently to 
4 


PROJECTIVE ORNAMENT 


appear, the conventionalization of our native grains, 
fruits and flowers, would undoubtedly introduce a 
note of fresh beauty and ap- 
propriateness into our archi- 
tecture. Teachers of design 
might put the problem of such 
conventionalizations before 
their pupils to their advantage, 
and to the advancement of art. 
There is, however, one diffi- 
culty that presents itself. By 
reason of scientific agriculture, 
intensive cultivation under 
glass, and because of the ease 
and freedom of present-day 
transportation, vegetation in 
civilized countries has lost 
much of its local character 
and significance. Corn, buck- 
.wheat, cotton, tobacco, though native to America, 
are less distinctively American than they once were. 
Moreover, dwellers in cities, where for the most part 
the giant flora of architecture lifts its skyscraping 
heads, know nothing of buckwheat except in pan- 
cakes, of cotton except as cloth or in the bale. Corn 
in the can is more familiar to them than corn on the 
cob, and not one smoker in ten would recognize 
tobacco as it grows in the fields. Our divorce from 
nature has become so complete that we no longer 
dwell in the old-time intimate communion with her 
visible forms. 


Pentahedroids: Tesseracts 


PROJECTIVE ORNAMERT 


GEOMETRY 


There remains at least one other possibility, and 
it is that upon which we shall now concentrate all 
our attention, for it seems indeed an open door. 
Geometry and number are at the root of every kind 
of formal beauty. That the tapestry of nature is 
woven on a mathematical framework is known to 
every sincere student. As Emerson says, “Nature 
geometrizes . . . moon, plant, gas, crystal, are con- 
crete geometry and number.” 
Art is nature selected, ar- 
ranged, sublimated, triply re- 
fined, but still nature, how- 
ever refracted in and by con- 
sciousness. If art is a higher 
power of nature, the former 
must needs submit itself to 
mathematical analysis too. 
The larger aspect of this whole 
matter—the various vistas 
that the application of geom- 
etry to design opens up—has 
been treated by the author in 
a previous volume*. Narrow- 
ed down to the subject of 
ornament, our question is, 
what promise does geometry 
hold of a new ornamental mode? 


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In the past, geometry has given birth to many 
characteristic and consistent systems of ornamenta- 
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*The Beautiful Necessity. 


mmeOoyeCclIVE ORNAMENT 


birth to many more. Much of Hindu, Chinese, and 
Japanese ornament was derived from geometry, yet 
these all differ from one another, and from Moorish 
ornament, which owes its origin to the same source. 
Gothic tracery, from Perpendicular to Flamboyant, 
is nothing but a system of straight lines, circles, 
and the intersecting arcs of circles, variously ar- 
ranged and combined. ‘The interesting development 
of ornament in Germany which has taken place of 
late years, contains few elements other than the 
square and the circle, the parallelogram and the 
ellipse. It is a remarkable fact that ornamentation, 
in its primitive manifestations, is geometrical rather 
than naturalistic, though the geometrical source is 
the more abstract and purely intellectual of the two. 
Is not this a point in its favor? The great war 
undoubtedly ends an era: “the old order changeth.” 
Our task is to create the art of the future: let us 
ney draw our inspiration from the deepest, purest 
well. 


Geometry is an inexhaustible well of formal 
beauty from which to fill our bucket; but before the 
draught is fit for use it should be examined, analyzed, 
and filtered through the consciousness of the artist. 


If with the zeal of the convert we set at once to 
work with IT square and compass to devise a new 
system of ornament from geometry, we shall proba- 
bly end where we began. Let us, therefore, by a 
purely intellectual process of analysis and selection, 
try to discover some system of geometrical forms 
and configurations which shall yield that new orna- 
mental mode of which we are in search. 


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II 


ORNAMENT AND PSYCHOLOGY 


Ornament is the outgrowth of no practical necessity, but of a striving 
toward beauty. Our zeal for efficiency has resulted in a corresponding 
aesthetic infertility. Signs are not lacking that consciousness is NOW 
looking in a new direction—away from the contemplation of the facts 
of materiality towards the mysteries of the supersensuous life. This 
transfer of attention should give birth to a new aesthetic, expres- 
sive of the changing psychological mood. The new direction of con- 
sciousness is well suggested in the phrase, The Fourth Dimension of 
Space, and the decorative motifs of the new aesthetic may appropri- 
ately be sought in four-dimensional geometry. 


THE ORNAMENTAL MODE AND THE PSYCHOLOGICAL 
MOOD 


RCHITECTURAL forms and features, such as 
the column, the lintel, the arch, the vault, are 
the outgrowth of structural necessity, but this is not 
true of ornament. Ornament develops not from the 
need and the power to build, but from the need and 
the power to beautify. Arising from a psychological 
impulse rather than from a physical necessity, it re- 
flects the national and racial consciousness. ‘To such 
a degree is this true that any mutilated and time- 
worn fragment out of the great past when art was a 
language can without difficulty be assigned its place 
and period. Granted a dependence of the ornamental 
mode upon the psychological mood, our first business 
is to discover what that mood may be. 
A great change has come over the collective 
consciousness: we are turning from the accumula- 
9 


PROJECTIVE ORNAMENT 


tion of facts to the contemplation of mysteries. 
Science is discovering infirmities in the very founda- 
tions of knowledge. Mathematics, through the 
questioning of certain postu- 
lates accepted as axiomatic 
for thousands of years, is 
concerning itself with prob- 
lems not alone of one-, two-, 
and three-, but of n-dimen- 
sional spaces. Psychology, 
no longer content with super- 
ficial manifestations, is plung- 
ing deeper and deeper into 
the examination of the sub- 
conscious mind. Philosophy, 
despairing of translating life 
by the rational method, in 
terms of inertia, is attempting 
to apprehend the universal 
flux by the aid of intuition. 
Religion is abandoning its 
man-made moralities of a superior prudence in favor 
of a quest for that mystical experience which fore- 
goes all to gain all. In brief, there is a renascence 
of wonder; and art must attune itself to this new 
key-note of the modern world. 


Icositetrahedroid 


THE FOURTH DIMENSION 


To express our sense of all this Newness many 

phrases have been invented. Of these the Fourth 

Dimension has obtained a currency quite outside the 

domain of mathematics, where it originated, and is 

frequently used as a synonym for what is new and 
10 


PeeyenG ll VE ORNAMENT 


strange. But a sure intuition lies behind this loose 
use of a loose phrase—the perception, namely, that 
consciousness is moving in a new direction; that it is 
glimpsing vistas which it must needs explore. 


Here, then, is the hint we have been seeking: 
consciousness is moving towards the conquest of a 
new space; ornament must indicate this movement 
of consciousness; geometry is the field in which we 
have staked out our particular claim. It follows, 
therefore, that in the soil of the geometry of four 
dimensions we should plant our metaphysical spade. 


The fourth dimension may 
be roughly defined as a direc- 
tion at right angles to every 
known direction. It is a 
hyperspace related to our 
space of three dimensions as 
the surface of a solid is re- 
lated to its volume; it is the 
withinness of the within, the 
outside of externality. 


“But this thou must not think to find 
With eyes of body but of mind.” 


We cannot point to it, we 
cannot picture it, though 
every point is the beginning 
of a pathway out of and 
into It. 


Double Prisms 


FOUR-DIMENSIONAL GEOMETRY 


However little the mathematician may be prepared 
to grant the physical reality of hyperspace—or, more 
1] 


PROJECTIVE ORNAM Ee 


properly, the hyperdimensionality of matter—its 
mathematical reality he would never call in question. 
Our plane and solid geometries are but the beginnings 
of this science. Four-dimen- 
sional geometry is far more 
extensive than three-dimen- 
sional. The numberof figures, 
and their variety, increases 
more and more rapidly as we 
mount to higher and higher 
spaces, each space extending 
in a direction not existing in 
the next lower space. More- 
over, these figures of hyper- 
space, though they are un- 
known to the senses, are 
known to the mind in great 
minuteness of detail. 


To the artist the richness 
of the field is not of great im- 
portance. He need concern 
himself with only a few of the more elementary 
figures of four-dimensional geometry, and only the 
most cursory acquaintance with the mathematical 
concepts involved in this geometry will give him all 
the material he seeks. 


Base of Icosahedroid: Cubes 


In the ensuing exposition, the willfulness and im- 
patience of the artistic temperament towards every- 
thing it cannot turn to practical account will be 
indulged to the extent of omitting all explanations 
and speculations not strictly germane to the purely 
aesthetic aspect of the matter. To such readers as 
are disposed to dig deeper, however, the author’s 

12 


PROJECTIVE ORNAMENT 


A Primer of Higher Space may be found useful, and 
there is besides a literature upon the subject. 


If after reviewing this literature the reader is 
disposed to regard the fourth dimension as a mere 
mathematical convention, it matters not in the 
least, so long as he is able to make practical use of it. 
He may likewise, with equal justice, question the 
existence of minus quantities, for example, but they 
produce practical results. 

With this brief explanation the author now turns 
up his shovelful, leaving it to the discerning to 
determine whether it contains any gold. 


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THE KEY TO PROJECTIVE 
ORNAMENT 


The idea of a fourth dimension is in conformity with reason, however 
foreign to experience. By means of projective geometry it is possible 
to represent a polyhedron (a three-dimensional figure) in the two 
dimensions of a plane. By an extension of the same method it is no 
less possible to represent a polyhedroid (a four-dimensional figure). 
Such representations in plane projection of solids and hypersolids 
constitute the raw material of Projective Ornament. 


THE DEVELOPMENT OF THE EQUILATERAL TRIANGLE 
IN HIGHER SPACES 


‘THE concept of a fourth dimension is so simple that 
almost anyone can understand it if only he will 
not limit his thought of that which is possible by 
his opinion of that which is practicable. It is not 
reason, but experience, that balks at the idea of 
four mutually perpendicular directions. Grant, 
therefore, if only for the sake of intellectual adven- 
ture, that there is a direction towards which we 
cannot point, at right angles to every one of the so- 
called three dimensions of space, and then see where 
we are able to come out. 

It is possible to locate in a plane (a two-dimen- 
sional space) three points, and only three, whose 
mutual distances are equal. This mathematical fact 
finds graphic expression in the equilateral triangle. 
(A, Figure 1). 

In three-dimensional, or solid space, it is possible 
to add a point, and the mutual equal distances, six 

15 


PROJECTIVE ORNAMENT 


PLANE, PROJECTIONS OF CORRESPONDING FIGURES OF THREE: 
AND OF FOUR:DIMENSIONAL SPACE<—— 


LX & AIAN 


TETRAHEDRAL CELLS Of PENTAHEDROD'D” 


TETRAHEDRAL CELLS OF PENTAHEDROID'E” 


in number, between the four points, will be expressed 
by the edges of a regular tetrahedron whose vertices 
are the four points. But in order to represent this 
solid in a plane, we must have recourse to projective 
geometry. The most simple and obvious way to 
do this is to locate the fourth point in the center of the 
equilateral triangle and draw lines from this central 
point to the three vertices. Then we have a rep- 
resentation of a regular tetrahedron as seen di- 
rectly from above, the central point representing 
the apex opposite the base (B, Figure 1). But suppose 
we imagine the tetrahedron to be tilted far enough 
over for this upper apex to fall (in plane projection) 
outside of the equilateral triangle representing the 
base. In such a position the latter would foreshorten 
to an isosceles triangle, and at a certain stage of this 
motion the plane projection of the tetrahedron 
appears as a square, its every apex representing 
an apex of the tetrahedron,whose edges are repre- 
sented by the sides and diagonals of the square (C, 
16 


PROJECTIVE ORNAMENT 


Figure 1). In this representation, though the points 
are equidistant on a plane, as they are equidistant 
in solid space, the six lines are not of the same 
length, and the four triangles are no longer truly 
equilateral. But this is owing to the exigencies of 
representation on a surface. If we imagine that we 
are not looking az a plane figure, but zn#o a solid, the 
necessary corrections are made automatically by the 
mind, and we have no difficulty in identifying the 
figure as a tetrahedron. 


Now if we concede to space another independent 
direction, in that fourth dimension we can add 
another point equidistant from all four vertices of the 
tetrahedron. ‘The mutual distances between these 
five points will be ten in number and all equal. The 
hypersolid formed—a pentahedroid—will be bounded 
by five equal tetrahedrons in the same way that a 
tetrahedron is bounded by four equal equilateral 
triangles, and each of these by three equal lines. 
We cannot construct this figure, for to do so would 
require a space of four dimensions, but we can rep- 
resent it in plane projection, just as we are able to 
represent a tetrahedron. We have only to add 
another point and connect it by lines with every 
point representing an apex of the original tetrahe- 
dron (D, Figure 1); or according to our second 
method we can arrange five points in such fashion as 
to coincide with the vertices of a regular pentagon 
and connect every one with every other one by 
means of straight lines (E, Figure 1). In either case 
by convention we have a plane representation of a 
hypertetrahedron or pentahedroid. 

17 


PROJECTIVE ORNAMENT 


If we have really achieved the plane representa- 
tion of a pentahedroid, it should be easy to identify 
the projections of the five tetrahedral cells or bound- 
ing tetrahedrons, just as we are able to identify the 
four equilateral sides of the tetrahedron in plane pro- 
jection. We find that it is possible to do this. For 
convenience of identification, these are separately 
shown. By dint of continued gazing at this pentagon 
circumscribing a five-pointed star, and by trying to 
recognize all its intricate inter-relations, we may come 
finally to the feeling that it is not merely a figure on a 
plane, but that it represents a hypersolid of hyper- 
space, related to the tetrahedron as that is related 
to the triangle. 


THE CORRESPONDING HIGHER DEVELOPMENTS OF THE 
SQUARE 


Let us next consider the series beginning with the 
square. The cube may be conceived of as developed 
by the movement of a 
TESSERACT GENERATION AND | square in a direction at 
fs PROJECTION right angles to its two 


dimensions, a distance 

equal to the length of 

La one of its sides. The 
direction of this move- 

TESSERACT@HvERcux| Ment can be represented 

CUBE on a plane anywhere 

we wish. Suppose we 
“ establish it as diagonal- 

ly downward andtotheright. The resultant figure is 
a cube in isometric perspective, for each of the four 


18 


PROJECTIVE ORNAMENT 


points has traced out a line, and each line has devel- 
oped a (foreshortened) square (Figure 2). The mind 
easily identifies the figure as a cube, notwithstanding 
the fact that the sides are not 
all squares, that the angles 
are not all equal, and that 
the edges are not all mutually 
perpendicular. 

Next let us, in thought, 
develop a hypercube, or tes- 
seract. To do this it will be 
necessary to conceive of a 
cube as moving into the fourth 
dimension a distance equal to 
the length of one of its sides. 
For plane representation we 
can, as before, assume this 
direction to be anywhere we 
like. Let it be diagonally 
downward, to the left. In 
this position we draw a second 
cube, to represent the first 
at the end of its motion into 


the fourth dimension. And : 
because each point has traced 


out a line, each line a square, and each square a 
cube, we must connect by lines all the vertices of 
the first cube with the corresponding vertices of the 
second. The resultant figure will be a perspective 
of a tesseract, or rather the perspective of a perspec- 
tive, for it is a two-dimensional representation of 
a three-dimensional representation of a four-dimen- 
sional form (Figure 2). 


THE EIGHT CUBES 
OF A TESSERACT 


SNK 
ACG 


19 


PROJECTIVE ORNAMENT 


If we have achieved 
the plane projection of 
: a tesseract we should be 


able to identify the 

LNINT NESTS S eight cubes by whee it 
TRIANGLES is bounds an pa at 

es ASIAN Sie tt Balt the beginning and end o 
LNDNINIAAAGAACA the iaoteel ar the six 
TETRAHEDRON developed by the move- 

Fed SAMs WE OS ment of the six faces of 
LBP LEAT PLBR OY the cube into four- 
PN NY NY |sdimensional space. We 
bGAA ey find that we can do this. 

For convenience of iden- 
tification the eight cubes 
are separately shown in 

Figure 3. 


SQUARES 
TE 


TRUTH TO THE MIND IS 
BEAUTY TO THE EYE. 


Ornament is largely a 
matter of the arrange- 
ment and repetition ofa 
few well chosen motifs. 
The basis of ornament 
is geometry. If we 
arrange these various 
geometrical figures in sequence and in groups we 
have the rudiments of ornament (Figure 4). Al- 
though all these are plane figures, there is this im- 
portant difference between them: the triangle and the 
square speak to the mind only in terms of two dimen- 
20 


WX 


PeeorerarlVyE ORNAMENT 


sions; the plane representations of the tetrahedron 
and the cube portray certain relations in solid space, 
while those of the pentahedroid and the tesseract 
portray relations peculiar to four-dimensional space. 
It will be observed that the 
decorative value of the figures 
increases as they proceed from 
space to space: the higher- 
dimensional developments are 
more beautiful and carry a 
greater weight of meaning. 
This accords well with the 
dictum, “Beauty is Truth; 
Truth, Beauty.” 


The above exercises consti- 

tute the only clue needed to 
understand the system of orna- 
ment here illustrated. Every 
symmetrical plane figure has 
its three-dimensional correla- 
tive, to which it is relatedas Terie poems 
a boundary or a cross-section. 
These solids may in turn be conceived of as boundaries 
or cross-sections of corresponding figures in four- 
dimensional space. The plane projections of these 
hypersolids are the motifs mainly used in Projective 
Ornament. 


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IV 
THREE REGULAR POLYHEDROIDS 


The paradoxes of four-dimensional geometry are best understood by 
referring them to the corresponding truisms of plane and of solid 
geometry. This may profitably be done in the case of the pentahe- 
droid, the tesseract, and the 16-hedroid, the four-fold figures of most 
use in Projective Ornament. In the plane representation of four-fold 
figures for decorative purposes certain conventions should be observed, 
conventions which, though they serve aesthetic ends, find justification 
in optical and physical laws. 


TWO-, THREE-, AND FOUR-FOLD FIGURES 


oe most effective method for a novice to approach 
an understanding of any four-dimensional figure 
can becompared tothe athletic exercise called the hop, 
skip and jump. In this the cumulative impetus 
given by the hop and the skip 1s concentrated and 
expended in the supreme effort of the jump. The 
jump into the fourth dimension is best prepared for, 
in any given case, by a preliminary hop in plane space, 
and a skip in solid space. 

In the following cursory consideration of the three 
simplest regular polyhedroids of four-dimensional 
space let us apply this method. Even at the risk of 
wearisome reiteration let us resolve the paradoxes 
of hyperspace by referring them to the truisms of 
lower spaces. 

A regular polygon—a two-fold figure—consists of 
equal straight lines so joined as to enclose symmetri- 
cally a portion of plane space. A regular polyhedron 
a three-fold figure—consists of a number of equal 

23 


PROJECTIVE ORNAM ESS 


regular polygons, together with their interiors, the 
polygons being joined by their edges so as to enclose 
symmetrically a portion of solid space. A regular 
polyhedroid consists of anum- 
ber of equal regular poly- 
hedrons, together with their 
interiors, the polyhedrons be- 
ing joined by their faces so 
as to enclose symmetrically a 
portion of hyperspace. 


In the foregoing chapter 
we have considered the two 
simplest regular polyhedroids: 
the regular pentahedroid, or 
hypertetrahedron, and the 
tesseract, or hypercube. To 
these let us now add the 
hexadekahedroid, or 16-hed- 
roid, bounded by 16-tetrahed- 
rons. These regular hyper- 
Octahedrons: Tetrahedrons solids are of such importance 
in Projective Ornament that their elements should 
be familiar, and their construction understood. 


THE PENTAHEDROID 


A regular pentahedroid is a regular figure of four- 
dimensional space bounded by five regular tetrahe- 
drons: it has five vertices, ten edges, ten faces, and 
five cells. | 

If we take an equilateral triangle and draw a line 
through its center perpendicular to its plane, every 
point of this line will be equidistant from the three 

24 


BeeOyeClTIVE ORNAMENT 


vertices of the triangle, and if we take for a fourth 
vertex that point on this line whose distance from 
the three vertices is equal to one of the sides of the 
triangle, we have then atetra- 
hedron in which the edges are 
all equal. 


If through the center of 
this regular tetrahedron we 
could draw a line perpendicu- 
lar to its hyperplane every 
point of this line would be 
similarly, as above, equidis- 
tant from the four vertices of 
the tetrahedron, and we could 
take for a fifth vertex a point 
at a distance from the four 
vertices equal to one of the 
edges of the tetrahedron. We 
would have then a penta- 
hedroid in which the ten 
edges would all be equal. 
All the parts of any one kind—face angles, dihedral 
angles, faces, etc.—would be equal; for the penta- 
hedroid is congruent to itself in sixty different ways 
and can be made to coincide with itself, any part 
coinciding with any other part of the same kind. 


Tetrahedrons: Icosahedrons 


As every regular polyhedroid can be inscribed 
in a hypersphere in the same way that a regular 
polygon can be inscribed in a circle, and every re- 
gular polyhedron in a sphere, the pentahedroid is 
most truly represented in plane projection as in- 
scribed within a circle representing this hypersphere. 
Radii perpendicular to the cells of the pentahedroid 

25 


PROJECTIVE ORNAMENT 


meet the hypersphere in 
five points which are the 
vertices of a second regular 
pentahedroid symmetric- 
ally situated to the first 
with respect to the center, 
and therefore equal to the 
first. Representing these 
vertices by equidistant in- 
termediate points on the 
circle circumscribing the 
pentahedroid and complet- 
ing the figure, we have a 
graphic representation of 


this fact (Figure 5). These 


PLANE PROJECTION OF 
TWO SYMMETRICALLY 
PLACED PEN'TAHEDROIDS' 
IN A HYPER SPHERE 


intersecting pentahedroids inscribed within a hyper- 
sphere have their analogue in plane space in two 
symmetrically intersecting equilateral triangles in- 
scribed within a circle, and in solid space in two 
symmetrical intersecting tetrahedrons inscribed with- 


in a sphere (Figure 6). 


THE TESSERACT 
The tesseract, or hypercube, 


lee ut is a regular figure of four- 
TERE ACIRCLE dimensional space having 


RON | eight cubical cells, twenty- 


four square faces, (each a 
common face of two cubes), 


Lip 
NV, thirty-two equal edges, and 


sixteen vertices. It con- 
tains four axes lying in lines 
which also form a rectangu- 


6 lar system. 


26 


PROJECTIVE ORNAM EW 


| CORRESPONDING PRO- 
| JECTIONS OF CUBE, AND 


TESSERACT bee 
A b 


In order to comprehend 
the tesseract in plane repre- 
resentation, let us first con- 
sider the corresponding 
plane representation of the 
cube. In parallel perspec- 
tive a cube appears as a 
square inside of another 
square, with oblique lines 
connecting the four vertices 
(A, Figure 7). By reason of 
our tactile and visual ex- 
perience, the inner and 
smaller square is thought 
of as the same size as the 
outer and larger, and the 
four intermediate quadri- 
lateral figures are thought 


of as squares also. If the cube is shown not in 
parallel, but oblique perspective, the mind easily 
identifies the two figures (B, Figure 7). 


These two ways of representing a cube in plane 
space may be followed in the case of the tesseract 
also (A’ and B’, Figure 7). We can think of the first 


as representing the ap- 
pearance of the tesse- 
ract as we look down 
into it, and the second 
as we stand a little to 
one side. In each case it 
is possible to identify the 
eight cubes whose in- 
teriors form the cells of 


GENERATION OF TESSERACT 


28 


Meee ye Cl IVE ORNAMENT 


the tesseract. The fact that 


they are not cubes except by ieee ard se iy 
convention is owing to the 

Eeecnvics of representation: ag LZ 
in four-dimensional space the , es 


cells are perfect cubes, and EQS Lip 


are correlated into a figure 


whose four dimensions are WS 
all equal. EWS SOS 
In order to familiarize vip ye, 
ourselves with this, for our Ze 
purposes the most impor- 


tant of all four-fold figures, ry 
let us again consider the Kd —YYj 
manner of its generation, be- \S TO pe 
ginning with the point. Let SES 
the point A, Figure 8, move 
to the right, terminating with \ We 
the point B. Next let the QR / 
\ ‘Un 


line AB move downward a 
distance equal to its length, 
tracing out the square AD. 


NY) 


(ZS 


THE TESSERACT IN THREE DIFFERENT ASPECTS 


PROJECTIVE ORNGM Es 


This square shall now move backward the same 
distance, generating the (stretched out) cube A H. 
And now, having exhausted the three mutually 
perpendicular directions of solid space, and undaunted 
by the physical impracticability of the thing, let this 
cube move off in a direction perpendicular to its 
every dimension (the fourth dimension) represented 
by the arrow. This will generate the tesseract Al. 
It will be found to contain eight cubical cells. For 
convenience of identification these are shown in 
Figure 9. Other aspects of the tesseract are shown 
in Figure 10; and in 

GENERATION OF TESSERACT| Figure 11 it is shown 
with an intermediate or 
cross-sectional square in 
each of the cubes, which 
square in the tesseract 
becomes an intermediate 
cube. Whenever, in the 
figure, we have three 
squares in the same 
straight line, we know 
that we have a cube. 
There are eight of these 
groups of three, the cubi- 
cal cells of the tesseract. 
If instead of represent- 
ing the fourth direction 
outside the generating 
cube we choose to con- 
ceive of it as inward, 
the resultant figure is 
that shown at the bot- 


30 


PROJECTIVE ORNAMENT 


tom of Figure 11, the innermost of these cubes cor- 
responding with the furthermost of the upper figure. 


THE 16-HEDROID 


After the pentahedroid or hypertetrahedron, and 
the tesseract or hypercube, already considered, we 
have as the next regular polyhedroid the hexadeka- 
hedroid, or, more briefly, the 16-hedroid. 


If we lay off a given distance in both directions 
on each of four mutually perpendicular lines inter- 
secting at a point, the eight points so obtained are 
the vertices of a regular 
polyhedroid which has four 
diagonals along the four |THE HEXADEKAHEDROID 
given lines. This is the 16- 
hedroid. It has, as the 
name implies, sixteen cells, 
(each a tetrahedron), thirty- 
two triangular faces, (each 
face common to two tetra- 
hedrons), twenty-four edges, 
and eight vertices. 

Figure 12 represents its 
projection upon a plane. 
The sixteen cells are ABCD, A’B’C’D’, AB’C’D’, 
Mpc. ABCD, A’BC’D, ABC’D, A‘B’CD’, 
pee b O'D),. ABC’D’, A’B/CD, A’BC’D, 
meow ABCD’, AB/C’D. The accented letters 
are the antipodes of the unaccented ones. Figure 13 
represents another plane projection of this poly- 
hedroid. 


31 


PROJECTIVE ORNGH 


THE DECORATIVE VALUE OF THESE FIGURES 


As this is a handbook for artists and not a geome- 
trical treatise, the description of regular polyhedroids 
need not be carried further than this. The reader 
who is ambitious to continue, from the 24-hedroid 
even unto the 600-hedroid, is referred to the geo- 
metry of four dimensions; upon this he can exer- 
cise his mind and experience for himself the stern 
joy of the conquest of new spaces. But the designer 
has already, in the penta- 
hedroid, the hypercube, and 
the hexadekahedroid, ample 
material on which to exer- 
cise his skill. It should be 
remembered that just as in 
plane geometry a regular 
polygon can always be in- 
scribed in a circle, and in 
geometry of three dimen- 
13 sions a regular polyhedron 
can always be inscribed in 
a sphere, so in four-dimensional geometry every 
regular polyhedroid can be inscribed in a hyper- 
sphere. In plane projection this hypersphere would 
be represented by a circle circumscribing the plane 
figure representing the polyhedroid. 


Almost any random arrangement on the page of 
these three hypersolids¥in plane projection will 
serve to indicate what largess of beauty is here— 
they are like cut jewels, like flowers, and like frost. 
Combined symmetrically they form patterns of 
endless variety. 


THE HEXADEKAHE DROID 


32 


MerewinGl tl VE ORNAMENT 


THE CONVENTIONS EMPLOYED IN THEIR 
REPRESENTATION 


There is a reason why the plane projections of 
hypersolids are shown as transparent. Our senses 
operate two-dimensionally—that is, we see and 
contact only surfaces. Were our sense mechanism 
truly three-dimensional, we should have X-ray 
vision, and the surfaces of solids would offer no re- 
sistance to the touch. In dealing with four-dimen- 
sional space we are at liberty to imagine ourselves in 
full possession of this augmented power of sight and 
touch. The mind having ascended into the fourth 
dimension, there would follow a _ corresponding 
augmentation on the part of the senses, by reason of 
which the interiors of solids would be as open as are 
the interiors of plane figures. 


There is justification also for the attenuation of 
all lines towards their center. It is in obedience to 
the optical law that when the light is behind an 
object it so impinges upon the intercepting object 
as to produce the effect of a 
OPTICAL EFFECTS | thinning towards the center. 
The actual form of the bars 
of a leaded glass window, for 
example, is as shown in A, 


‘A Figure 14, but their optical 
LIGHT effect when seen against the 
our S light is as in B. Because in 


X-ray vision some substances 

are Opaque, and some trans- 

lucent, we are at liberty to 

attribute opacity to any part 
33 


OPAQUE CENTER, 


PROJECTIVE ORNAMENT 


we please, and thus to add a new factor of variation 
as in C. We are also at liberty to stretch, twist or 
shear the figures in any manner we like. By the 
use of tones, of color, or by mitigating the crystalline 
rigidity of the figures through their combination with 
floral forms, we can create a new ornamental mode 
well adapted to the needs of today. 


eB) 
| at 


&, 
ra 


bx 


ora 


] 
J 


KA 


BINDING ATESSERACT OF STRETCHED 3-CUBES 


34 


7 | 


: on 


Wh 


WW, en if: 
| Oe 


NAAR 


a 


Pe 


Ed bo 


AVAD : 


xl aN 


LIA HY BF 
Ll ee NG 5 NN 
Od Ks 

ASO” 7K 
~ we Si N Uh 


SN be 


WE 


AN: va 
: NAN 


ZING , 


VIX Xi 


ae 


ay 


ay ie WA yi 


V. 


FOLDING DOWN 


Regular polyhedroids of four-dimensional space may be unfolded in 
three-dimensional space, and these again unfolded in a space of two- 
dimensions; or, contrariwise, they may be built up by assembling the 
regular polyhedrons which compose them. In this way new and valu- 
able decorative material is obtained. 


ANOTHER METHOD OF REPRESENTING THE HIGHER 
IN THE LOWER 


‘THE perspective method is not the only one 

whereby four-fold figures may be represented in 
three-dimensional and in two-dimensional space. 
Polyhedroids may be conceived of as cut apart along 
certain planes, and folded down into three-dimen- 
sional space in a manner analogous to that by which 
a cardboard box may be cut along certain of its 
edges and folded down into a plane. As the bounda- 
ries of a polyhedroid are polyhedrons, an unfolded 
polyhedroid will consist of a number of related 
polyhedrons. ‘These can in turn be unfolded, and 
the aggregation of polygons—each a plane boundary 
of the solid boundary of a hypersolid—will represent 
a four-fold figure unfolded in a space of two dimen- 
sions. 

An unfolded cube becomes a cruciform plane 
figure, made up of six squares, each one a boundary 
of the cube (A, Figure 15). Similarly, if we imagine 
a tesseract to be unfolded, its eight cubical cells will 
occupy three-dimensional space in the shape of a 
double-armed cross (B, Figure 15). In four-dimen- 

37 


PROJECTIVE ORNAMENT 


sional space these cubes can be turned in upon one 
another to form a symmetrical figure just as in 
three-dimensional space the six squares can be re- 
united to form a cube. 


A regular tetrahedron unfolded yields an equilat- 
eral triangle enclosed by three other equilateral 
triangles (C, Figure 
15). Similarly, an un- 
folded pentahedroid, 
or hyper tetrahedron, 
would consist of a cen- 
tral tetrahedron with 
four others resting on 
its four faces (D, Figure 
15). The pentahedroid 
could be re-formed by 
turning these towards 
one another in four- 
dimensional space, 
until they came com- 
pletely together again. 


A regular triangular 
prism unfolded yields 
three parallelograms, 

15 its sides; and two 

equilateral triangles, 

its ends (E, Figure 15). Similarly, a regular hyper- 
prism would unfold into four equal and similar 
triangular prisms and two tetrahedrons (F, Figure 
15). In four-dimensional space we could turn these 
prisms around the faces of the tetrahedron upon 
which they rest and the other tetrahedron around 
the face by which it is attached to one of the prisms, 

38 


FOLDED-DOWN FIGURES OF HIGHER SPACE 


mer ol lV E ORNAMENT 


and bring them all together, each prism with a 
lateral face resting upon a lateral face of each of the 
others, and each of the four faces of the second 
tetrahedron resting upon one of the prisms. This 
could be done without separating any of the figures, 
or distorting them in any way, and the figure thus 
folded up would then enclose completely a portion 
of four-dimensional space. 


THE POLYHEDRAL BOUNDARIES OF FOUR-DIMENSIONAL 
REGULAR ANGLES 


A regular angle for any dimensional space is one all 
of whose boundaries are the same inform and magni- 
tude. The summits of all regular figures in any 
space form regular angles since the distribution of 
their boundaries is sym- 
metrical and equal. G 


and H, Figure 16, repre- | BOUNDARIES OF REGULAR, 
sent respectively the en OFS AND CF'4 SPACE 


summits, one in each 
figure, of the tetrahedron | OO 
and the cube, with the 
two-dimensional bound- 


- . BOUNDARIES OF THE SUMMITS OF #& 
aries of the summit | tr et eae vei ea 


spread out symmetri- 
cally in a plane. The 
boundaries of the sum- 
mits of a four-dimension- G H’ 
al figure being solids, G’ | pounpartes oF THE SUMMITS OFA 
BeimerepresentTespec= ee eC 
tively the summits, one | DIMENSIONAL, $pack 

in each figure, of the 
higher correlatives of the 16 

39 


OUT F OMIM TICALLY IN PLANE SPACE 


PROJECTIVE ORNAMENT 


tetrahedron and the cube—the pentahedroid and the 
tesseract—spread out in three-dimensional space. 
That is, they represent, in three-dimensional per- 
spective, the symmetrical arrangement of the four 
boundaries of regular four-dimensional angles. In 
four-dimensional space the faces of those figures 
which lie adjacent to the common vertex are brought 
into coincidence, just as in three-dimensional space 
the edges of the triangles and squares adjacent to 
the common vertex are brought into coincidence, 
orn the summits of the tetrahedron and the 
cube. 


THE CONSTRUCTION OF THE 24-HEDROID 


It is possible to build up any regular polyhedroid by 
putting together a set of polyhedrons. We take 
them in succession in such order that each is joined 
to those already taken by a set of polygons like the 
incomplete polyhedron. 

Take the case of the four-fold icositetrahedroid or 
24-hedroid. I, Figure 17, shows a summit with six 
octahedral boundaries arranged about it symmetric- 
ally in three-dimensional space. Conceive I to be 
transported into four-dimensional space, and the 
interstices between the adjacent triangular faces to 
be closed up by joining those faces two and two; the 
figure assumes a form whose projection is represented 
in J with dotted lines omitted. Adjust to this figure 
twelve other octahedrons in a symmetrical manner; 
three of these octahedrons are represented by the 
dotted lines of J. Again, close up the interstices 
between the adjacent faces; the outline of the figure 
assumes a form whose projection is represented in K. 

40 


eH 
=; 3 Tamar 


ANAL 


Na 


EVN ia 
By 14 


) 


@ |}: e 
——— 
2 
~—_" = 


JU) Rg 
p's 
(0; 
1 4p. 


PROJECTIVE ORNAME 


Now conceive this figure to be turned inside out. 
There will be left in the middle of the figure a vacant 
space of exactly the form of J with the dotted lines 
omitted (L, Figure 17): 
such a group of six 
octahedronsis therefore 
required to complete 
the four-fold figure. By 
counting it is found that 
all the constituent octa- 
hedral summits of the 
four-fold figure are filled 
to saturation, and that 
the figure is in other 
respects complete and 
regular. The number of 
octahedral boundaries or cells is twenty-four; of 
summits, twenty-four; of triangular faces, ninety- 
six; of edges, ninety-six. 


CONSTRUCTION OF A 24 -HEDROID 


ae, 

: 

A 
] Nose 
nd K <) 


17 


TESSERACT SECTIONS 


In the same way that it is easy to conceive all 
regular polygons as two-dimensional boundaries or 
cross-sections of regular polyhedrons, it is possible, 
though not so easy, to conceive of these same polygons 
as boundaries or cross-sections of corresponding 
polyhedroids. 

The various figures are represented in perspective 
projection, but they may be unfolded, after the 
manner of the cardboard box. If this be done the 
bounding polygons will be free from the distortions 
incident to perspective representation, but the result 

42 


Peo CLIVE ORNAMENT 


REGULAR ICOSAHEDRON UNFOLDED IN A PLANE 


Ee 


OF ICOS. | OF ICOsAHEDEGN AHEDRON;S; ONE! INSIDE THE OTHER. 
18 


in most cases is the monotonous and uninteresting 

repetition of units (Figure 18). What we require 

for amanicnt is Bicates 

contrast and variety o 

saint form, and this may be 

DOWN AND ae IN A PLANE obtained without going 

farther than the won- 

der-box of the tesseract 
itself. 

There are certain 
interesting polyhe- 
droids embedded, as it 
were, in the tesseract. 
Such are the tetratesse- 
ract, and the octatesse- 
act. Lhis: lastis: ob-= 
tained by cutting off 
every corner of the 
tesseract Just as an Oc- 
tahedron 1s left if every 
corner of a cube is cut 
off. Three such poly- 


PROJECTIVE ORNAMENT 


hedral sections of a tesseract, unfolded, repeated, and 

arranged symmetrically with relation to one another, 
roduce the highly decorative pattern shown in 
igure: 19. 


jie 


Z 


y__\ 
\__/ 


Tt 


dg 


N 
Bm FN 
ua SS See ee ee eee eee 


aaa 


BINDING: FOUR TESSERACTS AND FOUR CUBES 


44 


! h 


| a 
i 


We 


D 


[E 


| 


/ 
i} 


E | 
Jes] 
SS 


VI 


MAGIC LINES IN MAGIC SQUARES 


The numerical harmony inherent in magic squares finds graphic expression 
in the magic lines which may be traced in them. These lines, trans- 
lated into ornament, yield patterns often of amazing richness and 
variety, beyond the power of the unaided aesthetic sense to compass. 
Magic lines have relations to spaces higher than a plane—they, too, 
are Projective Ornament. 


THE HISTORY OF MAGIC SQUARES 


AiMost everyone knows what a magic square Is. 

Briefly, it is a numerical acrostic, an arrangement 
of numbers in the form of a square, which, when added 
in vertical and horizontal rows and along the diago- 
nals, yield the same sum. Magic squares are of 
Eastern and ancient origin. There is a magic square 
of 4 carved in Sanskrit characters on the gate of 
the fort at Gwalior, in India (Figure 20). Engraved 
on stone and metal, magic squares are worn at the 
present day in the East as talismans or amulets. 
They are known to have occupied the attention 
of Mediaeval philosophers, astrologers, and mystics. 
Albrecht Direr introduced what is perhaps the most 
remarkable of all magic squares into his etching 
Melancholia (Figure 21). ‘Today they find place in 
the puzzle departments of magazines. Their laws 
and formulas have engaged the serious attention of 
eminent mathematicians, and the discovery of so- 
called magical relations between numbers, not alone 

47 , 


PROJECTIVE ORNAMEae 


in squares, but in cubes and hyper-cubes, is one of 
the recreations of the science of mathematics.* 


The artist, impatient of concept, but questing the 


AHINDU-SQUARE 


as[io]se 
haul] 27 
noe 


beautiful, will care little about 
the mathematical aspect of the 
matter, but it should interest him 
to know that the magic lines of 
magic squares are rich in decora- 
tive possibilities. 

A magic line is that endless 
line formed by following the 
numbers of a magic square in 
their natural sequence from cell 
to cell and returning to the point 
of departure. Because most 
magic squares are developed by 
arranging the numbers in their 
natural order in the form of a 
square and then subjecting them 
to certain rotations, the whole 
thing may be compared to the 
formation of string figures—the 
cat’s cradle of one’s childhood— 
in which a loop of string is made 
to assume various intricate and 
often amazing patterns—magic 
lines in space. 


*See Philip Henry Wynne’s Magic Tesseract in the author’s Primer of 


Higher Space. 


48 


memoyeCcTIVE ORNAMENT 


THEIR FORMATION 


Without going at all 
deeply into the arcana 
of the subject it will 
not be amiss to suggest 
one of the methods of 
magic square forma- 
tion by the simplest 
possible example, the 
magic square of 3. 
Arrange the digits in 
sequence in three hori- 
zontal lines, and re- 
late them to the cells 
of a square as shown 
in Figure 22. This will 
leave four cells empty 
and four numbers outside the perimeter. Dispose these 
numbers, not in the empty cells which they adjoin, 
but in the ones opposite; in other words, rotate the 
outside numbers in a direction at right angles 
to the plane of the paper, about the lines which 


FORMATION OF THE MAGIC SQUARE OF THREE 


22 


49 


PROJECTIVE ORNAA ESS 


MAGIC LINES IN MAGIC SQUARES 


47] 10 | 25| | 49] 2 | 9] 6 | 
2 63] 45| 2 | 62] S50] 3 | 


raf sfer ae] 1/2] 7 | 

exla[a ale [oe | 

re [i [25 |0[ 8 [4*] 8 [> 

afslals[a[a]4] [as] s9] a] a] 6 |x| | 

EIEYED COENEN 
CAEN EN a Ea Ea 

MAGIC SQUARE! OF 4.» MAGIC SQUARE OF 7 CHESS-BOARD 2ATH OF KNIGHT 


THE MAGIC LINE INA 1C SQUARE’ Is DISCOV 
ERED By Tec N¢ THe NUMERAL IN se 


ORDER F CELL, TO CELL, AND BACK TO THE 
BEGINNING NUMBER 
ewe ‘@ 


MAGIC LINE CF3 


Ri 


MAGIC LINE OF 4 MAGIC LINE OF 7 


severally bound the central cell. By this operation 
each outside number will fall in its proper place. 
These rotations are indicated by dotted lines. The 
result is the magic square of 3. Each line, in each 
of the two dimensions of the square, adds to 15, 
and the two diagonals yield the same sum. 


Now with a pencil, using a free-hand curve, 
follow the numbers in their order from 1 to 9 and 
back again to 1. The result is the magic line of the 

50 


PROJECTIVE ORNAMENT 


magic square of 3 (Figure 22). We have here a 
configuration of great beauty and interest, readily 


translatable into orna- 
ment. As the number 
of magic squares is 
practically infinite, and 
as each containsa magic 
line, here is a rich field 
for the designer, even 
though not all magic 
lines lend them selves 
to decorative treat- 
ment. Figures 23 and 
24, show some of them 
which do so lend them- 
selves, and Figures 25, 
26 and 27 show the 
translation of a few of 
these into ornament. 


eae ah © SQUARES 


MAGIC aii 5 


MAGIC LINFON4 MAGIC LINE O'S 


24 


THE KNIGHT’S TOUR 


It is a common feat of chess players to make the 
tour of the board by the knight’s move (two squares 
forward and one to right or left), starting at any 


“a a MAGIC LINES 


25 


square, touching at each 
square once, and returning to 
the point of departure. 
Keller, the magician, intro- 
duced this trick into his per- 
formance, permitting any 
member of the audience to 
designate the initial square. 


51 


PROJECTIVE ORNAYE ee 


It is a simple feat of 
mnemonics. ‘The per- 
former must remember 
64 numbers in their or- 
der, the sequence which 
yields the magic line in 
the magic square of 8. 
The plotting of this line 
is shown in Figure 23; 
its decorative applica- 
tion in the binding of 
The Beautiful Necessity. 
Euler, the great mathe- 
matician, constructed 
knight’s move squares 
of 5 and of 6, having 
peculiar properties. In 
26 one diagram of Figure 
28 the natural numbers 
show the path of a knight moving in such a manner 
that the sum of the pairs of numbers opposite to 
and equidistant from the middle figure is its double. 
In the other diagram the knight returns to its 
starting cell in such a manner that 
the difference between the pairs of 
numbers opposite to and equidis- 
tant from the middle point is 18. 


PATTERN FROM MAGIC SQUAEES 


CLINE OF 3 
aes 


PATH TRACED BY THE KNIGHT IN MAKING 
WHAT Is KNOWN AS THE KNIGHT'S TOURS 


INTERLACES 


Figure 28 shows interlaces derived 

from these two magic squares. 

They so resemble the braided bands 

found on Celtic crosses that one 
52 


Nave 04 CA IIPAS 
an 2 0.25 0 Ow 
oO” .“@.. 
Tere ek 


eee? Breasts, q 


oe oct tos! 


~Or1 etbeaeenseee ‘i; ©” 4 


~~ e 
yO ye OR 


Q ‘ 
PISO PS, 


PROJECTIVE ORNAMENT 


FULERS KNIGHT S-TOUR SQUARES 
| pola Ts [oes] 
[7 | 6 | 29 [20] 5 [| 
fs [3s| [27 


ref 92 far 
roofs] [a 


MAGIC LINE 
POM EULER 


su. 
Albrecht Durer, whose ac- 
quaintance with magic squares 
is a matter of record, is known 
to have expended a part of his 
inventive genius in designing 
interlacing knots. Leonardo 
da Vinci also amused himself 
in this way. The element of 
the mystic and mysterious 
entered into the genius of both 
these masters of the Renais- 
sance. One wonders if this 
may not have been due to 
some secret afhliation with an 
occult fraternity of adepts, 
whose existence and claims to 
the possession of extraordinary 


knowledge and power have 
54 


28 


naturally wonders if 
their unknown and ad- 
mirable artists may 
not have possessed the 
secret of deriving orna- 
ment from magic nu- 
merical arrangements, 
for these arrangements 
are not limited to the 
square, but embrace 
polygons of every des- 
cription. Here is an- 
other curious fact in 
this connection: 


PATTERNS FROM 
EULER'S KNIGHTS 
MOVE SQUARES 


<< = 
CREED 


Peon Cli VE ORNAMENT 


been the subject of much debate. Were these knots 
of theirs not only ornaments, but symbols—password 
and counter-sign pointing to knowledge not possess- 
ed by the generality of men? 


These patterns show forth in graphic form the 
symphonic harmony which abides in mathematics, 
a fact of sweeping significance, inasmuch as it 
involves the philosophical problem of the world- 
order. [he same order that prevails in these figures 
permeates the universe; through them one may 
sense ae cosmic har- 
mony of the spheres, 
just as it is possible to fy [is [is [| 
hear the ocean in a 5 | 


shell. 


MAGIC SQUARE AND 
CUBE OF 4. MAGIC 


THE PROJECTED 
MAGIC LINE 


In answer to any 
question which may 
arise in the mind of 
the reader as to the 
relevancy of magic 
squares to the subject 
of Projective Orna- 
ment, it may be stated 
that magic lines are 
Projective Ornament 
in a very strict sense. 
These lines, though 
figures on a_ plane, 
represent an extension 


PROJECTIVE ORNAMENT 


tion at right angles to the plane, and they have rela- 
tions to the third and higher dimensions. As this is a 
fact of considerable interest and importance, the 
attempt will be made to carry its demonstration at 
least far enough to assure the reader of its sub- 
stantial truth. Let us examine the three-dimensional 
aspect of the magic line in a magic square of 4. 


Figure 30 represents one of the most remarkable 
magic squares. Each horizontal, each vertical and 
each diagonal column adds 34. ‘The four corner 
cells add 34, and the four central cells add 34. The two 
middle cells of the top row add 34 with the two 
middle cells of the bottom row. The middle cells 
of the right and left columns similarly add 34. Go 
round the square clock-wise; the first cell beyond 
the first corner, plus the first beyond the second 
corner, plus the third, plus the fourth, equals 34. 
Take any number at random, find the three other 
numbers corresponding to it in any manner that 
respects symmetrically two dimensions, and the 
sum of the numbers is 34. 


In Figure 30 is also represented the magic cube 
of 4. It is made up of 64 cubical cells, each con- 
taining one of the numbers from 1 to 64, inclusive. 
This cube can be sliced into four vertical sections 
from left to right, or it can be separated into four 
other vertical sections by cutting planes perpendi- 
cular to the edge A B, proceeding from front to back, 
or the four sections may be horizontal, made by 
planes perpendicular to AD. 

Now each of these twelve sections presents a 
magic square in which each row and each column 
adds 130. The diagonals of these squares do not 

56 


Pewee oll) E ORNAMENT 


S*. 


OFA 


ry 


Cen 


BINDING: THE KNIGHTS TOUR (MAGIC LINE CF 8;SQUARE) 


add 130, but the four diagonals of the cube do add 
130. The essential correspondence of the magic 
square of 4 to the magic cube of 4 is clearly 
apparent. 

Now if we plot that portion of the magic line 
of the magic cube of 4 embraced by the numbers 
from 1 to 16 and compare it with the magic line 
of the magic square of 4, it is seen that the latter is 
a plane projection of the former. 

In other words, shut the four sections of the cube 
up so that the front section, A C, in 1-16 fits over 
the back section, 49-64; then using only the numbers 

57 


PROJECTIVE ORNAMENT 


1 to 16, they will be found to fall magically into the 
same places they occupy in the magic square of 4. 

Because all magic lines in magic squares have, in 
their corresponding cubes, this three-dimensional ex- 
tension, the patterns derived from magic squares 
come properly under the head of Projective Orna- 
ment. 


7A | 


[A 
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ASTISAISED 
EME ME 


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A MANMAE 
ISLS 
\*/ —<— 
KIA LAS 


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VN 
Zn 


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LAM SNZH SZ 


VY 


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WAN 


58 


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WSEAS 
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re Oa: TISORK 

Sie: me : 
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SS —w, —_ 


EIN SZ 
\ A a = (i: 
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Vee: a Cbs WAS aT 
a aN. “( YS le 
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eh 


WAZ aos 9] a PEEK = ee 


Vil 


A PHILOSOPHY OF ORNAMENT 


The language of form is a symbolical expression of the world order. 
This order presents itself to individual consciousness most movingly 
and dramatically under the guise of fate and of free-will. For these 
two the straight line and the curve are graphic expressions. An orna- 
mental mode should therefore embrace an intelligent and harmonious 
use of both. That Projective Ornament appears here so largely as a 
straight line system is because such a system is easier and more ele- 
mentary than the other, and because this is an elementary treatise— 
merely a point of departure for an all-embracing art of the future, 
only to be developed by the codperation of many minds. 


THE WORLD ORDER AND THE WORD ORDER 


PROJECTED solids and hypersolids, unfolded 
figures, magic lines in magic squares, these and 
similar translations of the truths of number into 
graphic form, are the words and syllables of the new 
ornamental mode. But we shall fail to develop a 
form language, eloquent and compelling, if we pre- 
occupy ourselves solely with sources—the mere 
lexicography of ornament. There is a grammar 
and a rhetoric to be mastered as well. The words 
are not enough, there remains the problem of the 
word order. 

Now the problem of the word order is the analogue 
of the problem of the world order. ‘The sublime 
function of true art is to shadow forth the world 
order through any frail and fragmentary thing a 
man may make with his hands, so that the great 
thing can be sensed in the little, the permanent in 

61 


PROJECTIVE ORNAM ES 


the transitory, as the sun, for 
instance, is imaged in a dande- 
lion, or a solar system in 
summer moths circling about 
a flame. 

The world order and the 
word order alike obey the law 
of polar opposites. The hard 
and sibilant in sound, the 
rigid and flowing in form, 
correspond to opposite 
powers: the former to that 
kind, igneous, masculine, 
which resists, and the latter 
to the aqueous, feminine type 
which prevails by yielding; 
Cubes: Line in Magic Square of 3 the first made the granite 

hills, the second, the fertile 
valleys. For these great opposites there are a 
thousand symbols: the cliff, the cloud; the oak, the 
vine—nature’s “inevitable duality.” One term 
corresponds to fate, destiny, and the other to free- 
will, forever forced to adjust itself to destiny. Each 
individual life, be it a Narcissus flower or a Napoleon, 
is the resultant of these two forces. The expansion 
of that life in space or on the field of action is deter- 
mined by what we name its “star”. In the case of 
the flower this is its invisible geometrical pattern to 
which the unfolding of every leaf and petal must 
conform; in the case of the man it is his destiny— 
his horoscope—the character with which he was born. 


62 


Poowenc li VE ORNAMENT 


FATE AND FREE-WILL AND THEIR SYMBOLS 


Here we have one of those universal truths, fixed 
from the foundation of the world. Fate decrees— 
“Thus far shalt thou go and no farther.” Free-will 
whispers—“ Within these limits thou art free.” 
Music figures these two admonitions of the spirit in 
the key, the beat, the movement, which correspond 
to destiny; and in the melody, which with all its 
freedom conforms to the key, obeys the beat, and 
comes to its appointed end in the return of the 
dominant to the tonic at the end of the passage. To 
symbolize the same two elements in ornament, 
what is for the first more fitting than figures of geo- 
metry, because they are absolute and inexorable; 
and for the second, than the fecund and free-spreading 
forms of vegetable life? 

Whether or not we choose 
to impute to geometrical and 
to floral forms the symbolical 
meaning here assigned them, 
we cannot fail to recognize 
these two elements in orna- 
ment, and a _ corresponding 
relation between them. There 
is the fixed frame or barrier, 
and there is the free-growing 
arabesque whose vigor faints 
against the crystalline rigidity 
of the frame—the diminishing 
energy returning upon itself 
in exquisite curves and spirals, 
like a wave from the face of a 
Line in Magic Squareofg _—iliff. In the language of orna- 

63 


PROJECTIVE ORNAMEe 


WY ment, here is an expression of 
os oe the highest spiritual truth— 


fate and free-will in perfect 
reconcilement. If from this 


point of view we consider even 
so hackneyed a thing as a 
Corinthian capital, the droop 
of the acanthus leaf where 
it meets the abacus becomes 


eloquent of that submission, 
after a life of effort, to a 
destiny beyond our failing 


# 
gt 


Tesseracts 


energy to overpass. This ex- 
quisite acquiescence, expressed 
thus in terms of form, is 
capable of affecting the emo- 
tions as music does— 


‘That strain again, it hath a dying fall.” 


It is the beautiful end to tragedy, summed up in 
Hamlet’s— 


“But let it be.—Horatio, I am dead.” 


POLARITY 


To create a new ornamental mode, we should con- 
ceive of ornament in this spirit, not as mere rhythmic 
space subdivision and flower conventionalization, 
but as symbology, most pregnant and profound. 
We must believe that form can teach as eloquently 
as the spoken word. 

The artist is not committed to a slavish fidelity 
to the forms of nature. God of his own self-created 

64 


PROJECTIVE ORNAMENT 


laws of nature he must be 
the world might be born, fell 
dering of a force into two Tesseracts 

We begin to learn this law almost at birth; youth | 
lesson. One Montessori exercise for very young 
those forms that are angular, like the tetrahedron 
world, should set himself a similar task. Time and 


world, he may fashion for it ‘Yj 
bound, for God himself, it has 
been said, is subject to the law 
of God. © @ 
asunder into man and wife.” 
Science says the same thing 
opposed activities striving for 
reunion, is a characteristic of all of the phenomena 
and maiden are learning it when they fall in love 
with one another, and philosophers, when adolescent 
children consists in providing them two boxes and 
a number of different geometrical solids made of 
and the cube, and in the other those that are smooth 
to the touch, like the egg and the sphere. The artist, 
space are his two boxes; his assemblage of figures, 
all of the contents of consciousness and of the world. 


a flora all its own; but by the 

What is this law? “Male 
and female created he them,”’ 
Genesis makes answer;and the 
Upanishads — “‘ Brahma, that © 
when it declares that the sun- : 
of nature, from magnet and crystal to man himself. 
fires die in the grate, are still engaged upon the same 
wood, with instructions to put together in one box 
a child more knowing, in the schoolroom of the 

65 


PROJECTIVE ORNAMENT 


SPACE AND TIME: THE FIELD AND THE FRAME 


Now the characteristic of time is succession; in time 
alone one thing follows another in endless sequence. 
The unique characteristic of space is simultaneity, 
for in space alone everything exists at once. In 
classifying the arts, for example, music would go 
into the time box, for it is in time alone, being 
successive; architecture, on the other hand, would 
go into the space box. Yet because nothing is pure, 
so to speak, architecture has something of the ele- 
ment of succession, and music of simultaneousness. 
An arcade or a colonnade may be spoken of as 
successive; while a musical chord, consisting of 
several notes sounded together, is simultaneous. 

The same thing holds true throughout nature. 
The time element and the space element everywhere 
appear, either explicitly or implicitly, the first as 
succession, the second as simultaneity. 

In ornament we have the field and the frame, 
and the unfolding of living forms in space within 
some fixed time cycle may be thought of as symbo- 
lized by a foliated field and a geometrical frame or 
border. In the field, the units will be disposed with 
relation to points and radiating lines, implying the 
simultaneity of space, and in the border they will 
be arranged sequentially, implying the succession of 
time (Figure 31). Seeking greater interest, subtlety, 
and variety, we have, in the projected ‘plane re- 
presentations of symmetrical three-fold and four- 
fold solids, a frame rhythmically subdivided. These 
subdivisions of a frame may be taken to represent 
lesser time cycles within a greater, and the arabesque 
with which these spaces can be filled may be felt to 

| 66 


cant iN 
y N'4 S4 oe ws : 
? ey he Bot 
we N'Z Be 2 4 
i us eee oo : 
a4 CeKK 
is ; oi 


x4 'Y wy ee 4 : 
N46 ANG M8 ; 


\f 
ke is er 


RORORPREN vk 


= Pa aN - 
ee 
man Ni NZ: 
ae s< ik OE 


PROJECTIVE ORKGi 


symbolize the growth of 
a plant through succes- pe He forte, MBOL; 
siveseasons, or the SB) - 


WEES 
indi- SUCCESSIVE + (@ 
development of an indi Preach hi (OY : 
vidual in different incar- 
nations. 


A BOOK ALL BONES AND 
NO FLESH 


SIMULTANEOUS 


It is by artifices such as 
these that the world 
order gets itself external- \ RQ 
ized in forms and _ ar- SUCCESSIVE 
rangements which ex- 
press “‘the life movement 
of the spirit through 
the rhythm of things.” This is the very essence 
of art: first to perceive, and then to publish 
news from that nowhere of the world from which all 
things flow and to which all things return. It will 
be evident to the discerning reader from what has 
been said regarding the symbolic value of the straight 
line and the curve (the frame and the arabesque) 
that the whole subject of foliated or free-spreading 
ornament has received scant attention from the 
author. ‘This intentional concentration upon the 
straight line explains the poverty and hard monotony 
of many of the diagrams here presented. They are 
not so much ornament as the osseous framework of 
ornament. But by reason of our superficial manner 
of observing nature, our preoccupation with mere 
externals, we have lost our perception of her beauti- 
68 


31 


repel IVY k ORNAMENT 


ful bones—her geometry. When we have recovered 
that, the rest is easy. It has seemed best not to 
complicate the subject nor confuse the issue, by 
proceeding to show (as one might) the relation of 
floral forms to geometrical figures, for this is some- 
thing that every artist can look into for himself. 


69 


Sf 
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— 
ne 
D 
=! 


————— 
ee hk 


—— 


" 
4 
A 
" 
‘ i} 
Coe 
an aay ‘ 
’ ’ 
Ne i : 
\\ > ’ i 
iY 4 
= 
| 
> 
i 
| 


oo ma 


Vill 


THE USES OF PROJECTIVE 
ORNAMENT 


Projective Ornament, being directly derived from geometry, is universal 
in its nature. It is not a compendium of patterns, but a system for 
the creation of patterns. Its principles are simple and comprehensive 
and their application to particular problems stimulates and develops 
the aesthetic sense, the mind, and the imagination. 


THE FIELD AND FUNCTION OF PROJECTIVE ORNAMENT 


PROJECTIVE Ornament is that rhythmic sub- 
division of space expressed through the figures 
of Projective Geometry. As rhythmic space sub- 
division is of the very essence of ornament, Pro- 
jective Ornament possesses the element of univer- 
sality, though it lends itself to some uses more 
readily than to others. To those crafts which 
employ linear design, such as lace-work, lead-work, 
book-tooling, and the art of the jeweler, it is particu- 
larly well suited; with color it lends itself admirably 
to stained glass, textiles, and ceramics. On the 
other hand, it must be considerably modified to 
give to wrought iron an appropriate expression: 
its application to cast iron and wood-inlaying pre- 
sents fewer difficulties. Its three-dimensional, as 
well as its two-dimensional aspects, come into play 
in architecture, and from its many admirable geo- 
metrical forms there might be developed architectural 
detail pleasing alike to the mind and to the eye. A 
crying need of the time would thus be met. The drab 
71 


PROJECTIVE OKNAG 


aN BR RL Ae e monotony of broad cement 

ww, Catt “os surfaces could be relieved by 
a 2 oN Vx \ A means of incrusted ornament 
a: ow, in colored tiles arranged in 
yob,° ° 


Xi, <4! patterns developed by the 


i AS 


7 
a Nig " methods described. 


67,9 


Various applications of 
Projective Ornament to prac- 
tical problems are suggested 
in the page illustrations dis- 
persed throughout this vol- 
ume, but a careful study of 

Y WSS the text will be more profit- 

A “> able to the designer than any 
‘3 as SA copying of the designs. If 

evar pacenias the rationale of the system 
See ran is thoroughly grasped, a de- 
signer will no longer need to copy patterns, since 
he will have gained the power to create new ones for 
himself. To copy is the death of art. No worse 
fate could befall this book, or the person who would 
profit by it, than to use it merely as a book of patterns 
‘These should be looked upon only as illustrative of 
certain fundamental principles susceptible of endless 
application. Mr. Sullivan, from sad experience, 
predicted that the zeal of any converts that the 
book might make would be expended in sedulous 
imitation rather than in original creation. The 
author, however, takes a more hopeful view. 


HOW TO AWAKEN THE SLEEPING BEAUTY 


The principles here set forth are eminently com- 
municable and understandable. They present no 
72 


Peo ymeo lI YE ORNAMENT 


difficulty, even to an intelligent child. Indeed, the 
fashioning and folding up of elementary geometrical 
solids is a kindergarten exercise. The great impedi- 
ment to success in this field is a proud and sophisti- 
‘cated mind. Let the learner “become as a little 
child,’ therefore: let him at all times exercise him- 
self in Observational Geometry—that is, look 
for the simple geometrical forms and relations of 
the objects that come under his every-day notice. 
He should come to recognize that the myriad forms 
in the animal, vegetable, and 
: mineral kingdoms furnish an 
GING 7 } unending variety of symmetri- 
= fs Zar cal and complex geometric 
; forms which may be discovered 
and applied to his own prob- 
lems. This should create an 
appetite for the study of 
Formal Geometry. From that 
study a fresh apprehension of 
the beauty of arithmetical 
relations is sure to follow. 
Enamored of this beauty, the 
disciple will seek out the basic 
geometrical ground rhythms 
latent in nature and in human 
life. The development of 
Hexadekahedroids faculty will follow on the 
awakening of perception: the 

elements and relations grasped by the mind will 
externalize themselves in the work of the hand. 
Not content with the known and familiar space re- 
lationships, the student will essay to explore the 


73 


PROJECTIVE ORNAWEW 


field of hyperspace. But let him not seek to achieve 
results too easily and too quickly. In all his work he 
should follow an orderly sequence, quarrying his 
gold before refining it, and fashioning it to his 
uses only after it is refined: that is, he should 
endeavor to understand the figures before he draws 
them, and he should draw them as geometrical 
diagrams before he attempts to alter and combine 
them for decorative use. It is the author’s experience 
that they will require very little alteration; that 
they are in themselves decorative. The filling in of 
certain spaces for the purpose of achieving notan 
(contrast) is all that is usually required. This 
done, the application of color is the next step in the 
process: first comes line, then light and dark, and 
lastly color values. Such is the method of the 
Japanese, those masters of 
decorative design. 


THE ILLUSTRATIONS AND 
DIAGRAMS 


The black-and-white designs 
interspersed throughout the 
text represent Projective Orna- 
ment removed only one degree 
from geometrical diagrams, — 
yet they are seen to be highly 
decorative even in this form. 
At the pleasure of the designer 
they may be elongated, con- 
tracted, sheared, twisted, 
translated from straight lines 
74 


AG 
BY; 


We Si 


V7 
f AG 
<7 O== 
Zk Cres ey 


YAyp Sos EN 


Ze (CW 


Wa 
(GS h 
wa A\ 


PROJECTIVE ORNGA Ye 


into curves; and by subjecting them to these modi- 
fications their beauty is often augmented. Yet if 
their geometrical truth and integrity be too much 
tampered with, they will be found to have lost a 
certain precious quality. It would seem as though 
they were beautiful to the eye in proportion as they 


-|}- 
W771 


Neo, AL 
(4 \Y he 7 

“No ( 

2. 


Fd 1 Vi ee 1 ee ed 1 
: Av? Hl 
mars ae As bal 


kc 
Ly 


q 
ee 


NS VW 


N 
oe", 


( 


(xi »@ 
Iw @ 


= A 
SN74S2 72 NA] 


VAS 


76 


feel ly EF ORNAMENT 


are interesting to the mind. For the sake of variety 
the figures are presented in three different ways; 
that is, in the form of mons, borders, and fields— 
corresponding to the point, the line, and the plane. 
It is clear that all-over patterns quite as interesting 
as those shown may be formed by repeating some of 
the unit figures. With this scant alphabet it is 
possible to spell more words than one or two. 


Projective Ornament, derived as it is from Pro- 
jective Geometry, is a new utterance of the trans- 
cendental truth of things. Whatever of beauty the 
figures in this book show forth has its source, not 
in any aesthetic idiosyncracy of the illustrator, 
but in that world order which number and geometry 
represent. ‘hese figures illustrate anew the idea, 
old as philosophy itself, that all forms are projections 
on the lighted screen of a material universe of 
archetypal ideas: that all of animate creation is 
one vast moving picture of the play of the Cosmic 
Mind. With the falling away of all our sophistries, 
this great truth will again startle and console man- 
kind—that creation is beautiful and that it is 
necessitous, that the secret of beauty is necessity. 
*‘Let us build altars to the Beautiful Necessity.”’ 


CONCLUSION 


Emerson says, ‘‘ Perception makes. Perception has 

a destiny.” How can new beauty be born into the 

world except by the awakening of new percep- 

tion? Evolution is the master-key of modern 

science, but that very science ignores the evolution 

of consciousness—of perception. ‘This it treats as 
77 


PROJECTIVE ORN AWE 


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fixed, static. On the contrary, it is fluent, dynamic. 

Were it not so, there would be little hope of a new art. 

The modern mind has adventured far and fear- 

lessly in the new realms of thought opened up by 

research and discovery, but it has left no trail of 

beauty. ‘That it has not done so is the fault of the 
78 


PeeeCtiV E ORNAMENT 


artist, who has failed to interpret and portray the 
movement of the modern mind. Enamored of an 
outworn beauty, he has looked back, and like Lot’s 
- wife, he has become a pillar of salt. The outworn 
beauty is the beauty of mere appearances. The new 
beauty, which corresponds to the new knowledge, 
is the beauty of principles: not the world aspect, 
but the world order. The world order is most 
perfectly embodied in mathematics. This fact is 
recognized in a practical way by the scientist, who 
increasingly invokes the aid of mathematics. It 
should be recognized by the artist, and he should 
invoke the aid of mathematics too. 


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